lighting/shading i week 6, fri feb 12

University of British ColumbiaCPSC 314 Computer Graphics

Jan-Apr 2010

Tamara Munzner

http://www.ugrad.cs.ubc.ca/~cs314/Vjan2010

Lighting/Shading I

Week 6, Fri Feb 12

2

Correction: W2V vs. V2W

• MW2V=TR

• we derived position of camera in world• invert for world with respect to camera

• MV2W=(MW2V)-1=R-1T-1

!

T =

1 0 0 ex

0 1 0 ey

0 0 1 ez

0 0 0 1

"

#

$ $ $ $

%

&

' ' ' '

!

R =

ux vx wx 0

uy vy wy 0

uz vz wz 0

0 0 0 1

"

#

$ $ $ $

%

&

' ' ' '

!

Mview2world

=

ux uy uz 0

vx vy vz 0

wx wy wz 0

0 0 0 1

"

#

$ $ $ $

%

&

' ' ' '

1 0 0 (ex0 1 0 (ey0 0 1 (ez0 0 0 1

"

#

$ $ $ $

%

&

' ' ' '

=

ux uy uz (e •u

vx vy vz (e • v

wx wy wz (e •w

0 0 0 1

"

#

$ $ $ $

%

&

' ' ' '

slide 38 week3.day3 (Fri Jan 22)

3

Correction: W2V vs. V2W

• MV2W=(MW2V)-1=R-1T-1

!

Mview2world

=

ux uy uz 0

vx vy vz 0

wx wy wz 0

0 0 0 1

"

#

$ $ $ $

%

&

' ' ' '

1 0 0 (ex0 1 0 (ey0 0 1 (ez0 0 0 1

"

#

$ $ $ $

%

&

' ' ' '

=

ux uy uz (e •u

vx vy vz (e • v

wx wy wz (e •w

0 0 0 1

"

#

$ $ $ $

%

&

' ' ' '

!

MV2W

=

ux uy uz "ex # ux + "ey # uy + "ez # uzvx vy vz "ex #vx + "ey #vy + "ez #vzwx wy wz "ex #wx + "ey #wy + "ez #wz

0 0 0 1

$

%

& & & &

'

(

) ) ) )

4

Correction: Perspective Derivation

!

x '

y '

z'

w'

"

#

$ $ $ $

%

&

' ' ' '

=

E 0 A 0

0 F B 0

0 0 C D

0 0 (1 0

"

#

$ $ $ $

%

&

' ' ' '

x

y

z

1

"

#

$ $ $ $

%

&

' ' ' '

!

y'= Fy + Bz, y'

w'=Fy + Bz

w', 1=

Fy + Bz

w', 1=

Fy + Bz

"z,

1= Fy

"z+ B

z

"z, 1= F

y

"z" B, 1= F

top

"("near)" B,

!

x'= Ex + Az

y'= Fy + Bz

z'= Cz + D

w'= "z

!

x = left " # x / # w = $1

x = right " # x / # w =1

y = top " # y / # w =1

y = bottom " # y / # w = $1

z = $near " # z / # w =1

z = $ far " # z / # w = $1

!

1= Ftop

near" B

z axis flip! slide 30 week4.day3 (Fri Jan 29)

5

News

• P2 due date extended to Tue Mar 2 5pm• V2W correction affects Q1 and thus cascades

to Q4-Q7• perspective correction affects Q8

• TA office hours in lab for P2/H2 questions Fri2-4 (Garrett)

PPT Fix: Basic Line Drawing

• assume• , slope

• one octant, other cases symmetric• how can we do this more quickly?

• goals• integer coordinates• thinnest line with no gaps

00

01

01 )()(

)(yxx

xx

yyy

bmxy

+!!

!=

+=

!

Line ( x0, y0, x1, y1)

begin

float dx, dy, x, y, slope ;

dx" x1 # x0;

dy" y1 # y0;

slope"dydx

;

y" y0

for x from x0 to x1 do

begin

PlotPixel ( x, Round (y) ) ;

y" y + slope ;

end ;

end ;!

0 <dydx

<1

!

x0

< x1

7

Clarification/Correction II: Midpoint• we're moving horizontally along x direction (first octant)

• only two choices: draw at current y value, or move up vertically toy+1?

• check if midpoint between two possible pixel centers above or below line• candidates

• top pixel: (x+1,y+1)• bottom pixel: (x+1, y)

• midpoint: (x+1, y+.5)• check if midpoint above or below line

• below: pick top pixel• above: pick bottom pixel

• other octants: different tests• octant II: y loop, check x left/right above: bottom pixel

below: top pixel

8

Review: Triangulating Polygons• simple convex polygons

• trivial to break into triangles• pick one vertex, draw lines to all others not

immediately adjacent• OpenGL supports automatically

• glBegin(GL_POLYGON) ... glEnd()

• concave or non-simple polygons• more effort to break into triangles• simple approach may not work• OpenGL can support at extra cost

• gluNewTess(), gluTessCallback(), ...

9

P

Review: Flood Fill

• simple algorithm• draw edges of polygon• use flood-fill to draw interior

10

PPT Fix: Flood Fill• draw edges• run:

• drawbacks?

!

FloodFill(Polygon P, int x, int y, Color C)

if not (OnBoundary(x,y,P) or Colored(x,y,C))

begin

PlotPixel(x,y,C);

FloodFill(P,x +1,y,C);

FloodFill(P,x,y +1,C);

FloodFill(P,x,y "1,C);

FloodFill(P,x "1,y,C);

end ;

11

Review: Scanline Algorithms

• scanline: a line of pixels in an image• set pixels inside polygon boundary along

horizontal lines one pixel apart vertically• parity test: draw pixel if edgecount is odd• optimization: only loop over axis-aligned

bounding box of xmin/xmax, ymin/ymax

1

2

3

4

5=0P

12

Review: Bilinear Interpolation

• interpolate quantity along L and R edges,as a function of y

• then interpolate quantity as a function of x

yy

P(x,y)P(x,y)

PP11

PP22

PP33

PPLL PPRR

13

1P

3P

2P

P

Review: Barycentric Coordinates• non-orthogonal coordinate system based on

triangle itself• origin: P1, basis vectors: (P2-P1) and (P3-P1)

γ=1

γ=0

β=1

β=0

α=1

α=0

P = P1 + β(P2-P1)+γ(P3-P1)P = (1-β−γ)P1 + βP2+γP3

P = αP1 + βP2+γP3

α + β+ γ = 10 <= α, β, γ <= 1 ((α,β,γα,β,γ) =) =

(0,1,0)(0,1,0)

((α,β,γα,β,γ) =) =(1,0,0)(1,0,0)

((α,β,γα,β,γ) =) =(0,0,1)(0,0,1)

14

Using Barycentric Coordinates• weighted combination of vertices• smooth mixing• speedup

• compute once per triangle

• demohttp://www.cut-the-knot.org/Curriculum/Geometry/Barycentric.shtml

1P

3P

2P

P

((α,β,γα,β,γ) =) =(1,0,0)(1,0,0)

((α,β,γα,β,γ) =) =(0,1,0)(0,1,0)

((α,β,γα,β,γ) =) =(0,0,1)(0,0,1) 5.0=!

1=!

0=!

321PPPP !+!+!= "#$

1,,0

1

!!

=++

"#$

"#$

““convex combination of pointsconvex combination of points””

for points inside triangle

15

Computing Barycentric Coordinates• 2D triangle area• half of parallelogram area

• from cross product

A = ΑP1 +ΑP2 +ΑP3

α = ΑP1 /Aβ = ΑP2 /Aγ = ΑP3 /A

3PA

1P

3P

2P

P

((α,β,γα,β,γ) =) =(1,0,0)(1,0,0)

((α,β,γα,β,γ) =) =(0,1,0)(0,1,0)

((α,β,γα,β,γ) =) =(0,0,1)(0,0,1) 2

PA

1PA

16

PP22

PP33

PP11

PPLL PPRRPPdd22 : d : d

11

3

21

12

21

2

3

21

12

21

1

23

21

12

)1(

)(

Pdd

dP

dd

d

Pdd

dP

dd

d

PPdd

dPP

L

++

+=

=+

++

!=

!+

+=

Deriving Barycentric From Bilinear

• from bilinear interpolation of point P onscanline

17

Deriving Barycentric From Bilineaer

• similarly

bb 11 :

b

:

b22

PP22

PP33

PP11


11

1

21

12

21

2

1

21

12

21

1

21

21

12

)1(

)(

Pbb

bP

bb

b

Pbb

bP

bb

b

PPbb

bPP

R

++

+=

=+

++

!=

!+

+=

18

bb 11 :

b

:

b22

PP22

PP33

PP11


11

3

21

1

2

21

2P

dd

dP

dd

dPL

++

+=

1

21

1

2

21

2P

bb

bP

bb

bPR

++

+=cc11: c: c22


• combining

• gives

!

P =c2

c1

+ c2

d2

d1

+ d2

P2

+d1

d1

+ d2

P3

"

# $

%

& ' +

c1

c1

+ c2

b2

b1

+ b2

P2

+b1

b1

+ b2

P1

"

# $

%

& '

!

P =c2

c1+ c

2

" PL

+c1

c1+ c

2

" PR

19


• thus P = αP1 + βP2 + γP3 with

• can verify barycentric properties

1,,0,1 !!=++ "#$"#$!

" =c1

c1+ c

2

b1

b1+ b

2

# =c2

c1+ c

2

d2

d1+ d

2

+c1

c1+ c

2

b2

b1+ b

2

$ =c2

c1+ c

2

d1

d1+ d

2

20

Lighting I

21

Rendering Pipeline

GeometryDatabase

Model/ViewTransform. Lighting Perspective

Transform. Clipping

ScanConversion

DepthTestTexturing Blending

Frame-buffer

22

Projective Rendering Pipeline

OCS - object/model coordinate system

WCS - world coordinate system

VCS - viewing/camera/eye coordinatesystem

CCS - clipping coordinate system

NDCS - normalized device coordinatesystem

DCS - device/display/screen coordinatesystem

OCSOCS O2WO2W VCSVCS

CCSCCS

NDCSNDCS

DCSDCS

modelingmodelingtransformationtransformation

viewingviewingtransformationtransformation

projectionprojectiontransformationtransformation

viewportviewporttransformationtransformation

perspectiveperspectivedividedivide

object world viewing

device

normalizeddevice

clipping

W2VW2V V2CV2C

N2DN2D

C2NC2N

WCSWCS

23

Goal• simulate interaction of light and objects• fast: fake it!

• approximate the look, ignore real physics• get the physics (more) right

• BRDFs: Bidirectional Reflection Distribution Functions• local model: interaction of each object with light• global model: interaction of objects with each other

24

Photorealistic Illumination

[[electricimageelectricimage.com].com]

•transport of energy from light sources to surfaces & points•global includes direct and indirect illumination – more later

Henrik Wann Henrik Wann JensenJensen

25

Illumination in the Pipeline• local illumination

• only models light arriving directly from lightsource

• no interreflections or shadows• can be added through tricks, multiple

rendering passes• light sources

• simple shapes• materials

• simple, non-physical reflection models

26

Light Sources• types of light sources• glLightfv(GL_LIGHT0,GL_POSITION,light[])

• directional/parallel lights• real-life example: sun• infinitely far source: homogeneous coord w=0

• point lights• same intensity in all directions

• spot lights• limited set of directions:

• point+direction+cutoff angle

!!!!

"

#

$$$$

%

&

0

z

y

x

!!!!

"

#

$$$$

%

&

1

z

y

x

27

Light Sources

• area lights• light sources with a finite area• more realistic model of many light sources• not available with projective rendering pipeline

(i.e., not available with OpenGL)

28

Light Sources

• ambient lights• no identifiable source or direction• hack for replacing true global illumination

• (diffuse interreflection: light bouncing off fromother objects)

29

Diffuse Interreflection

30

Ambient Light Sources

• scene lit only with an ambient light source

Light PositionNot Important

Viewer PositionNot Important

Surface AngleNot Important

31

Directional Light Sources

• scene lit with directional and ambient light

Light PositionNot Important

Viewer PositionNot Important

Surface AngleImportant

32

Point Light Sources

• scene lit with ambient and point light source

Light PositionImportant

Viewer PositionImportant

Surface AngleImportant

33

Light Sources• geometry: positions and directions• standard: world coordinate system

• effect: lights fixed wrt world geometry• demo:

http://www.xmission.com/~nate/tutors.html• alternative: camera coordinate system

• effect: lights attached to camera (car headlights)• points and directions undergo normal

model/view transformation• illumination calculations: camera coords

34

Types of Reflection• specular (a.k.a. mirror or regular) reflection causes

light to propagate without scattering.

• diffuse reflection sends light in all directions withequal energy.

• mixed reflection is a weightedcombination of specular and diffuse.

35

Specular Highlights

36

Types of Reflection

• retro-reflection occurs when incident energyreflects in directions close to the incidentdirection, for a wide range of incidentdirections.

• gloss is the property of a material surfacethat involves mixed reflection and isresponsible for the mirror like appearance ofrough surfaces.

37

Reflectance Distribution Model

• most surfaces exhibit complex reflectances• vary with incident and reflected directions.• model with combination

+ + =

specular + glossy + diffuse = reflectance distribution

38

Surface Roughness

• at a microscopic scale, allreal surfaces are rough

• cast shadows onthemselves

• “mask” reflected light:shadow shadow

Masked Light

39

Surface Roughness

• notice another effect of roughness:• each “microfacet” is treated as a perfect mirror.• incident light reflected in different directions by

different facets.• end result is mixed reflectance.

• smoother surfaces are more specular or glossy.• random distribution of facet normals results in diffuse

reflectance.

40

Physics of Diffuse Reflection

• ideal diffuse reflection• very rough surface at the microscopic level

• real-world example: chalk• microscopic variations mean incoming ray of

light equally likely to be reflected in anydirection over the hemisphere

• what does the reflected intensity depend on?

41

Lambert’s Cosine Law

• ideal diffuse surface reflectionthe energy reflected by a small portion of a surface from a light sourcein a given direction is proportional to the cosine of the angle betweenthat direction and the surface normal

• reflected intensity• independent of viewing direction• depends on surface orientation wrt light

• often called Lambertian surfaces

42

Lambert’s Law

intuitively: cross-sectional area ofthe “beam” intersecting an elementof surface area is smaller for greaterangles with the normal.

43

Computing Diffuse Reflection• depends on angle of incidence: angle between surfacenormal and incoming light• Idiffuse = kd Ilight cos θ

• in practice use vector arithmetic• Idiffuse = kd Ilight (n • l)

• always normalize vectors used in lighting!!!• n, l should be unit vectors

• scalar (B/W intensity) or 3-tuple or 4-tuple (color)• kd: diffuse coefficient, surface color• Ilight: incoming light intensity• Idiffuse: outgoing light intensity (for diffuse reflection)

nl

θ

44

Diffuse Lighting Examples

• Lambertian sphere from several lightingangles:

• need only consider angles from 0° to 90°• why?• demo: Brown exploratory on reflection• http://www.cs.brown.edu/exploratories/freeSoftware/repository/edu/brown/cs/ex

ploratories/applets/reflection2D/reflection_2d_java_browser.html

lighting/shading i week 6, fri feb 12

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